Okay, let's tackle this problem step by step. A worker earns $20,000 annually and deposits it at the end of each year into a savings account. By the end of the third year, he wants the total to be at least $66,200. We need to find the minimal compound interest rate required. Hmm, so this is a future value of an ordinary annuity problem. Let me recall the formula for that.

The future value of an ordinary annuity is calculated using the formula:
FV = PMT * [(1 + r)^n - 1] / r

Where:
- FV is the future value,
- PMT is the annual payment,
- r is the annual interest rate,
- n is the number of periods.

In this case, PMT is $20,000, n is 3 years, FV is $66,200. We need to solve for r. But since this equation is a bit complicated with r in both the exponent and the denominator, it might not be straightforward to solve algebraically. Maybe I need to use trial and error or some numerical method here.

Let me write down the equation again:
66,200 = 20,000 * [(1 + r)^3 - 1] / r

Divide both sides by 20,000:
66,200 / 20,000 = [(1 + r)^3 - 1] / r

Calculating the left side:
66,200 / 20,000 = 3.31

So,
3.31 = [(1 + r)^3 - 1] / r

Multiply both sides by r:
3.31r = (1 + r)^3 - 1

Bring all terms to one side:
(1 + r)^3 - 3.31r - 1 = 0

Let me expand (1 + r)^3:
1 + 3r + 3r^2 + r^3 - 3.31r - 1 = 0

Simplify the terms:
1 + 3r + 3r^2 + r^3 - 3.31r - 1 = 0

The 1 and -1 cancel out:
3r + 3r^2 + r^3 - 3.31r = 0

Combine like terms:
(3r - 3.31r) + 3r^2 + r^3 = 0
-0.31r + 3r^2 + r^3 = 0

Factor out an r:
r(-0.31 + 3r + r^2) = 0

So, r = 0 is a solution, but that's trivial and not applicable here. The other solutions come from:
r^2 + 3r - 0.31 = 0

Wait, but that seems off because when we factor out r, the remaining quadratic is r^2 + 3r - 0.31 = 0. Let me check my algebra again.

Wait, original equation after expanding:
(1 + r)^3 - 3.31r - 1 = 0

Expanding (1 + r)^3 gives 1 + 3r + 3r^2 + r^3. Subtract 1 and 3.31r:
1 + 3r + 3r^2 + r^3 - 1 - 3.31r = 0

Simplify: 3r - 3.31r = -0.31r, so:
-0.31r + 3r^2 + r^3 = 0

Which factors as r(r^2 + 3r - 0.31) = 0

Ah, so the quadratic equation is r^2 + 3r - 0.31 = 0. Let's solve that.

Using quadratic formula:
r = [-3 ± sqrt(9 + 1.24)] / 2

Calculate discriminant:
9 + 1.24 = 10.24

sqrt(10.24) = 3.2

So,
r = [-3 ± 3.2]/2

We have two solutions:
r = (-3 + 3.2)/2 = 0.2/2 = 0.1 or 10%

r = (-3 - 3.2)/2 = -6.2/2 = -3.1

Since interest rate can't be negative, r = 0.1 or 10%.

Wait a minute, but if I plug r = 10% into the original equation, does it satisfy?

Let's check:

FV = 20,000 * [(1 + 0.1)^3 - 1]/0.1

Calculate (1.1)^3 = 1.331

So, (1.331 - 1)/0.1 = 0.331/0.1 = 3.31

Multiply by 20,000: 20,000 * 3.31 = 66,200. Perfect, that matches.

But wait, the quadratic equation solution gave us exactly 10%, which is the answer. So, does that mean the minimal required interest rate is 10%? But wait, let me think again. The problem says "minimal compound interest rate". But if we found that at 10%, the future value is exactly 66,200, then any rate higher than 10% would result in a higher future value. Therefore, 10% is the minimal rate needed. If the rate were lower, the future value would be less than 66,200. So, 10% is indeed the minimal required.

But let me verify this with another method to ensure there's no mistake. Let's compute the future value at 10%:

Year 1: Deposit $20,000 at the end of year 1. It earns interest for 2 years.
Year 2: Deposit $20,000 at the end of year 2. It earns interest for 1 year.
Year 3: Deposit $20,000 at the end of year 3. No interest earned.

Calculations:

End of Year 1: $20,000 * 1.10 = $22,000
End of Year 2: $22,000 * 1.10 + $20,000 = $24,200 + $20,000 = $44,200
End of Year 3: $44,200 * 1.10 + $20,000 = $48,620 + $20,000 = $68,620

Wait, that's $68,620, which is more than $66,200. But according to the formula earlier, it's exactly $66,200. There's a discrepancy here. Wait, maybe I made a mistake in the calculation.

Wait, no. Let's recalculate step by step.

Wait, the formula gave FV = 20,000 * [(1.1)^3 - 1]/0.1

Which is 20,000 * (1.331 - 1)/0.1 = 20,000 * 0.331 / 0.1 = 20,000 * 3.31 = 66,200. So that's correct.

But when I computed year by year:

End of Year 1: $20,000 * 1.1 = $22,000

End of Year 2: $22,000 * 1.1 + $20,000 = $24,200 + $20,000 = $44,200

End of Year 3: $44,200 * 1.1 + $20,000 = $48,620 + $20,000 = $68,620

Wait, this is different. So which one is correct?

Wait, the formula is the future value of an ordinary annuity, which is the sum of each payment's future value. Let's compute it step by step:

Year 1 deposit: $20,000 earns interest for 2 years: 20,000*(1.1)^2 = 20,000*1.21 = $24,200

Year 2 deposit: $20,000 earns interest for 1 year: 20,000*1.1 = $22,000

Year 3 deposit: $20,000 earns interest for 0 years: $20,000

Total FV = 24,200 + 22,000 + 20,000 = $66,200. Oh! So that matches the formula. Wait, but when I did the step-by-step year by year, I got $68,620. That's because I was compounding each year's total, which is wrong. The correct way is to compound each deposit separately and then sum them.

So, the formula is correct. The step-by-step calculation should be:

At the end of each year, the amount is:

End of Year 1: 20,000*(1.1) = 22,000

End of Year 2: [22,000 + 20,000]*(1.1) = 42,000*1.1 = 46,200

Wait, hold on. Wait, no. Wait, the deposits are made at the end of each year, so the amount at the end of each year is:

End of Year 1: 20,000 (deposited) + interest on previous amount. But since it's the end of the year, the first deposit is at the end of year 1. So the amount at the end of year 1 is 20,000. Then, at the end of year 2, you add another 20,000, making total 40,000, and then that amount earns interest. Then at the end of year 3, you add another 20,000.

Wait, maybe I need to clarify the timeline.

Let's define t=0 as the present. The first deposit is at the end of year 1 (t=1), second at end of year 2 (t=2), third at end of year 3 (t=3). So, the amount at t=3 is:

20,000*(1 + r)^2 + 20,000*(1 + r) + 20,000.

Which is exactly the same as the future value of an ordinary annuity formula. So, that's why the formula is correct.

But when I calculated it step by step earlier, I think I messed up the timing. Let's do it again step by step.

Year 1:

Deposit: $20,000 at end of year 1. So at the end of year 1, the amount is $20,000.

Year 2:

Deposit another $20,000 at end of year 2. So at the end of year 2, the total amount is $20,000*(1 + r) + $20,000. Then, at the end of year 3, that amount is multiplied by (1 + r) again, and then another $20,000 is added.

Wait, no. Wait, each deposit is made at the end of each year. So, the first deposit is at the end of year 1, which then earns interest for 2 more years (years 2 and 3). The second deposit is at the end of year 2, earning interest for 1 year (year 3). The third deposit is at the end of year 3, earning no interest.

Therefore, the total FV is:

First deposit: 20,000*(1 + r)^2

Second deposit: 20,000*(1 + r)^1

Third deposit: 20,000*(1 + r)^0 = 20,000

Therefore, total FV = 20,000*(1 + r)^2 + 20,000*(1 + r) + 20,000

Which is the same as the future value of an ordinary annuity formula. So when we plug in r = 10%, we get:

20,000*(1.21) + 20,000*(1.1) + 20,000 = 24,200 + 22,000 + 20,000 = 66,200. Perfect, that's correct.

But when I did the step-by-step calculation earlier, I think I added the amounts incorrectly. Let me redo that.

If r = 10%, then:

End of Year 1: $20,000 (deposited) + interest from previous. But since it's the first deposit, only $20,000.

End of Year 2: $20,000 (from Year 1) * 1.1 = $22,000; then deposit another $20,000, so total $42,000.

End of Year 3: $42,000 * 1.1 = $46,200; deposit another $20,000, total $66,200.

Ah, there we go! So that matches the formula. So, step by step:

Year 1: $20,000. End of year 1.

Year 2: $20,000*(1.1) + $20,000 = $22,000 + $20,000 = $42,000.

Year 3: $42,000*(1.1) + $20,000 = $46,200 + $20,000 = $66,200.

So that's correct. Earlier, I made a mistake in the step-by-step calculation by not properly compounding each year's total. So, the formula is correct, and r = 10% is indeed the minimal rate.

But let me check with a slightly lower rate, say 9%, to see if the FV is less than 66,200.

Calculate FV at 9%:

First deposit: 20,000*(1.09)^2 = 20,000*1.1881 = 23,762

Second deposit: 20,000*(1.09) = 21,800

Third deposit: 20,000

Total FV: 23,762 + 21,800 + 20,000 = 65,562. Which is less than 66,200. So, 9% is too low.

Similarly, 10% gives 66,200, which is exactly the target. Therefore, 10% is the minimal required rate.

Wait, but just to be thorough, let's check 10.5% to see if maybe the minimal rate is slightly less than 10%. But according to the equation, the quadratic solution gave exactly 10%, which matches the exact FV. So, maybe 10% is the exact answer. Let's verify with 10.0001% just to check.

But before that, let me think again. The equation was solved as a quadratic, leading to r = 0.1. So, perhaps this is the exact solution. Since the quadratic equation resulted in r = 0.1, which is 10%, and substituting back gives the exact FV. Therefore, 10% is the minimal rate. There's no need for trial and error here because the equation yields an exact solution.

Therefore, the answer is 10%.

But wait, let me confirm once more. The formula solution gives r = 10%, and substituting back into the formula gives exactly 66,200. So, that's correct. The minimal rate is 10%.

So, the answer is $\boxed{10}$.
Therefore, the minimal compound interest rate that the savings account must provide is $\boxed{10}$.
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